A Comparative Study of Covariance and Precision Matrix ... · Motivation Setup Results Summary A...
Transcript of A Comparative Study of Covariance and Precision Matrix ... · Motivation Setup Results Summary A...
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MotivationSetup
ResultsSummary
A Comparative Study of Covariance and Precision
Matrix Estimators for Portfolio Selection
M. Senneret1 Y. Malevergne2,3 P. Abry4 G. Perrin1
L. Ja�rès1
1Vivienne Investissement2Université de Lyon - Université de Saint-Etienne
3EMLYON Business School4CNRS, ENS Lyon
AFFI, 2013
M. Senneret, Y. Malevergne, P. Abry, G. Perrin, L. Ja�rès Covariance and Precision for Portfolio Selection
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MotivationSetup
ResultsSummary
Outline
1 Motivation
2 Setup
Assets & Optimizations
Direct Estimates
Factor Models
Shrinkage
Sparsity
3 Results
Volatility Control
Sharpe Control
Concentration Control
M. Senneret, Y. Malevergne, P. Abry, G. Perrin, L. Ja�rès Covariance and Precision for Portfolio Selection
![Page 3: A Comparative Study of Covariance and Precision Matrix ... · Motivation Setup Results Summary A Comparative Study of Covariance and Precision Matrix Estimators for Portfolio Selection](https://reader034.fdocuments.in/reader034/viewer/2022043014/5fafca008eff9505ca481637/html5/thumbnails/3.jpg)
MotivationSetup
ResultsSummary
Estimation & Inversion
Covariance input is central to Mean Variance Portfolio (MVP)
Estimation of covariance su�ers from large estimation
�uctuations, even for large n (# of samples).
Mean Variance problem =⇒ inversion of covariance matrix.
Estimation error + numerical instability
=⇒ MVP≺ equally-weighted (EWP) - DeMiguel et al. (09)
Best use of known information (var/covar) ?
M. Senneret, Y. Malevergne, P. Abry, G. Perrin, L. Ja�rès Covariance and Precision for Portfolio Selection
![Page 4: A Comparative Study of Covariance and Precision Matrix ... · Motivation Setup Results Summary A Comparative Study of Covariance and Precision Matrix Estimators for Portfolio Selection](https://reader034.fdocuments.in/reader034/viewer/2022043014/5fafca008eff9505ca481637/html5/thumbnails/4.jpg)
MotivationSetup
ResultsSummary
Estimation & Inversion
Covariance input is central to Mean Variance Portfolio (MVP)
Estimation of covariance su�ers from large estimation
�uctuations, even for large n (# of samples).
Mean Variance problem =⇒ inversion of covariance matrix.
Estimation error + numerical instability
=⇒ MVP≺ equally-weighted (EWP) - DeMiguel et al. (09)
Best use of known information (var/covar) ?
M. Senneret, Y. Malevergne, P. Abry, G. Perrin, L. Ja�rès Covariance and Precision for Portfolio Selection
![Page 5: A Comparative Study of Covariance and Precision Matrix ... · Motivation Setup Results Summary A Comparative Study of Covariance and Precision Matrix Estimators for Portfolio Selection](https://reader034.fdocuments.in/reader034/viewer/2022043014/5fafca008eff9505ca481637/html5/thumbnails/5.jpg)
MotivationSetup
ResultsSummary
Estimation & Inversion
Covariance input is central to Mean Variance Portfolio (MVP)
Estimation of covariance su�ers from large estimation
�uctuations, even for large n (# of samples).
Mean Variance problem =⇒ inversion of covariance matrix.
Estimation error + numerical instability
=⇒ MVP≺ equally-weighted (EWP) - DeMiguel et al. (09)
Best use of known information (var/covar) ?
M. Senneret, Y. Malevergne, P. Abry, G. Perrin, L. Ja�rès Covariance and Precision for Portfolio Selection
![Page 6: A Comparative Study of Covariance and Precision Matrix ... · Motivation Setup Results Summary A Comparative Study of Covariance and Precision Matrix Estimators for Portfolio Selection](https://reader034.fdocuments.in/reader034/viewer/2022043014/5fafca008eff9505ca481637/html5/thumbnails/6.jpg)
MotivationSetup
ResultsSummary
Estimation & Inversion
Covariance input is central to Mean Variance Portfolio (MVP)
Estimation of covariance su�ers from large estimation
�uctuations, even for large n (# of samples).
Mean Variance problem =⇒ inversion of covariance matrix.
Estimation error + numerical instability
=⇒ MVP≺ equally-weighted (EWP) - DeMiguel et al. (09)
Best use of known information (var/covar) ?
M. Senneret, Y. Malevergne, P. Abry, G. Perrin, L. Ja�rès Covariance and Precision for Portfolio Selection
![Page 7: A Comparative Study of Covariance and Precision Matrix ... · Motivation Setup Results Summary A Comparative Study of Covariance and Precision Matrix Estimators for Portfolio Selection](https://reader034.fdocuments.in/reader034/viewer/2022043014/5fafca008eff9505ca481637/html5/thumbnails/7.jpg)
MotivationSetup
ResultsSummary
Estimation & Inversion
Covariance input is central to Mean Variance Portfolio (MVP)
Estimation of covariance su�ers from large estimation
�uctuations, even for large n (# of samples).
Mean Variance problem =⇒ inversion of covariance matrix.
Estimation error + numerical instability
=⇒ MVP≺ equally-weighted (EWP) - DeMiguel et al. (09)
Best use of known information (var/covar) ?
M. Senneret, Y. Malevergne, P. Abry, G. Perrin, L. Ja�rès Covariance and Precision for Portfolio Selection
![Page 8: A Comparative Study of Covariance and Precision Matrix ... · Motivation Setup Results Summary A Comparative Study of Covariance and Precision Matrix Estimators for Portfolio Selection](https://reader034.fdocuments.in/reader034/viewer/2022043014/5fafca008eff9505ca481637/html5/thumbnails/8.jpg)
MotivationSetup
ResultsSummary
Covariance or Precision?
Notations:
Precision matrix Θ
Covariance matrix Σ
with Θ = Σ−1
Θ is the input of interest
Σ̂−1: poor estimator of Θ.
Develop reliable estimators of Θ
Compare results.
M. Senneret, Y. Malevergne, P. Abry, G. Perrin, L. Ja�rès Covariance and Precision for Portfolio Selection
![Page 9: A Comparative Study of Covariance and Precision Matrix ... · Motivation Setup Results Summary A Comparative Study of Covariance and Precision Matrix Estimators for Portfolio Selection](https://reader034.fdocuments.in/reader034/viewer/2022043014/5fafca008eff9505ca481637/html5/thumbnails/9.jpg)
MotivationSetup
ResultsSummary
Covariance or Precision?
Notations:
Precision matrix Θ
Covariance matrix Σ
with Θ = Σ−1
Θ is the input of interest
Σ̂−1: poor estimator of Θ.
Develop reliable estimators of Θ
Compare results.
M. Senneret, Y. Malevergne, P. Abry, G. Perrin, L. Ja�rès Covariance and Precision for Portfolio Selection
![Page 10: A Comparative Study of Covariance and Precision Matrix ... · Motivation Setup Results Summary A Comparative Study of Covariance and Precision Matrix Estimators for Portfolio Selection](https://reader034.fdocuments.in/reader034/viewer/2022043014/5fafca008eff9505ca481637/html5/thumbnails/10.jpg)
MotivationSetup
ResultsSummary
Covariance or Precision?
Notations:
Precision matrix Θ
Covariance matrix Σ
with Θ = Σ−1
Θ is the input of interest
Σ̂−1: poor estimator of Θ.
Develop reliable estimators of Θ
Compare results.
M. Senneret, Y. Malevergne, P. Abry, G. Perrin, L. Ja�rès Covariance and Precision for Portfolio Selection
![Page 11: A Comparative Study of Covariance and Precision Matrix ... · Motivation Setup Results Summary A Comparative Study of Covariance and Precision Matrix Estimators for Portfolio Selection](https://reader034.fdocuments.in/reader034/viewer/2022043014/5fafca008eff9505ca481637/html5/thumbnails/11.jpg)
MotivationSetup
ResultsSummary
Covariance or Precision?
Notations:
Precision matrix Θ
Covariance matrix Σ
with Θ = Σ−1
Θ is the input of interest
Σ̂−1: poor estimator of Θ.
Develop reliable estimators of Θ
Compare results.
M. Senneret, Y. Malevergne, P. Abry, G. Perrin, L. Ja�rès Covariance and Precision for Portfolio Selection
![Page 12: A Comparative Study of Covariance and Precision Matrix ... · Motivation Setup Results Summary A Comparative Study of Covariance and Precision Matrix Estimators for Portfolio Selection](https://reader034.fdocuments.in/reader034/viewer/2022043014/5fafca008eff9505ca481637/html5/thumbnails/12.jpg)
MotivationSetup
ResultsSummary
Constraints as indirect regularization
Jagannathan & Ma (2003) : introduction of constraints =⇒better out-of-sample performance
investment restrictions act as shrinkage of Σ
How is this result related to the instability of Σ̂−1?
M. Senneret, Y. Malevergne, P. Abry, G. Perrin, L. Ja�rès Covariance and Precision for Portfolio Selection
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MotivationSetup
ResultsSummary
Constraints as indirect regularization
Jagannathan & Ma (2003) : introduction of constraints =⇒better out-of-sample performance
investment restrictions act as shrinkage of Σ
How is this result related to the instability of Σ̂−1?
M. Senneret, Y. Malevergne, P. Abry, G. Perrin, L. Ja�rès Covariance and Precision for Portfolio Selection
![Page 14: A Comparative Study of Covariance and Precision Matrix ... · Motivation Setup Results Summary A Comparative Study of Covariance and Precision Matrix Estimators for Portfolio Selection](https://reader034.fdocuments.in/reader034/viewer/2022043014/5fafca008eff9505ca481637/html5/thumbnails/14.jpg)
MotivationSetup
ResultsSummary
Constraints as indirect regularization
Jagannathan & Ma (2003) : introduction of constraints =⇒better out-of-sample performance
investment restrictions act as shrinkage of Σ
How is this result related to the instability of Σ̂−1?
M. Senneret, Y. Malevergne, P. Abry, G. Perrin, L. Ja�rès Covariance and Precision for Portfolio Selection
![Page 15: A Comparative Study of Covariance and Precision Matrix ... · Motivation Setup Results Summary A Comparative Study of Covariance and Precision Matrix Estimators for Portfolio Selection](https://reader034.fdocuments.in/reader034/viewer/2022043014/5fafca008eff9505ca481637/html5/thumbnails/15.jpg)
MotivationSetup
ResultsSummary
Assets & OptimizationsDirect EstimatesFactor ModelsShrinkageSparsity
Outline
1 Motivation
2 Setup
Assets & Optimizations
Direct Estimates
Factor Models
Shrinkage
Sparsity
3 Results
Volatility Control
Sharpe Control
Concentration Control
M. Senneret, Y. Malevergne, P. Abry, G. Perrin, L. Ja�rès Covariance and Precision for Portfolio Selection
![Page 16: A Comparative Study of Covariance and Precision Matrix ... · Motivation Setup Results Summary A Comparative Study of Covariance and Precision Matrix Estimators for Portfolio Selection](https://reader034.fdocuments.in/reader034/viewer/2022043014/5fafca008eff9505ca481637/html5/thumbnails/16.jpg)
MotivationSetup
ResultsSummary
Assets & OptimizationsDirect EstimatesFactor ModelsShrinkageSparsity
Assets & strategy
Universe: p = 211 european stocks of the STOXX R©Europe
600
Between December 14, 2001 and January 24, 2013
Covariance and precision estimated over 3 rolling windows:
n = 150, 200, 350
Every �ve days rebalancing
No fees
M. Senneret, Y. Malevergne, P. Abry, G. Perrin, L. Ja�rès Covariance and Precision for Portfolio Selection
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MotivationSetup
ResultsSummary
Assets & OptimizationsDirect EstimatesFactor ModelsShrinkageSparsity
Unconstrained Optimization
the global MV portfolio without restriction w∗ is given by
w∗ =Θ1p1′pΘ1p
Θ ? Σ̂−1 ?
n ≤ p ?
M. Senneret, Y. Malevergne, P. Abry, G. Perrin, L. Ja�rès Covariance and Precision for Portfolio Selection
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MotivationSetup
ResultsSummary
Assets & OptimizationsDirect EstimatesFactor ModelsShrinkageSparsity
Unconstrained Optimization
the global MV portfolio without restriction w∗ is given by
w∗ =Θ1p1′pΘ1p
Θ ? Σ̂−1 ?
n ≤ p ?
M. Senneret, Y. Malevergne, P. Abry, G. Perrin, L. Ja�rès Covariance and Precision for Portfolio Selection
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MotivationSetup
ResultsSummary
Assets & OptimizationsDirect EstimatesFactor ModelsShrinkageSparsity
Constrained Optimization
With short sales restrictions:
w∗ = argminw w ′Σw
s.t. 1′pw = 1
w ≥ 0
Lagrangian
L(w ,λ ,ν) = w ′Σw + λ (1′pw −1)−ν
′w ,
M. Senneret, Y. Malevergne, P. Abry, G. Perrin, L. Ja�rès Covariance and Precision for Portfolio Selection
![Page 20: A Comparative Study of Covariance and Precision Matrix ... · Motivation Setup Results Summary A Comparative Study of Covariance and Precision Matrix Estimators for Portfolio Selection](https://reader034.fdocuments.in/reader034/viewer/2022043014/5fafca008eff9505ca481637/html5/thumbnails/20.jpg)
MotivationSetup
ResultsSummary
Assets & OptimizationsDirect EstimatesFactor ModelsShrinkageSparsity
Outline
1 Motivation
2 Setup
Assets & Optimizations
Direct Estimates
Factor Models
Shrinkage
Sparsity
3 Results
Volatility Control
Sharpe Control
Concentration Control
M. Senneret, Y. Malevergne, P. Abry, G. Perrin, L. Ja�rès Covariance and Precision for Portfolio Selection
![Page 21: A Comparative Study of Covariance and Precision Matrix ... · Motivation Setup Results Summary A Comparative Study of Covariance and Precision Matrix Estimators for Portfolio Selection](https://reader034.fdocuments.in/reader034/viewer/2022043014/5fafca008eff9505ca481637/html5/thumbnails/21.jpg)
MotivationSetup
ResultsSummary
Assets & OptimizationsDirect EstimatesFactor ModelsShrinkageSparsity
Direct Estimates
if n < p , then Σ is not full rank =⇒ No inversion
if n ' p , then Σ is near criticical point =⇒Innaccuracy
Use Moore Penrose Inverse
Use diagonal matrix
Use Identity
M. Senneret, Y. Malevergne, P. Abry, G. Perrin, L. Ja�rès Covariance and Precision for Portfolio Selection
![Page 22: A Comparative Study of Covariance and Precision Matrix ... · Motivation Setup Results Summary A Comparative Study of Covariance and Precision Matrix Estimators for Portfolio Selection](https://reader034.fdocuments.in/reader034/viewer/2022043014/5fafca008eff9505ca481637/html5/thumbnails/22.jpg)
MotivationSetup
ResultsSummary
Assets & OptimizationsDirect EstimatesFactor ModelsShrinkageSparsity
Direct Estimates
if n < p , then Σ is not full rank =⇒ No inversion
if n ' p , then Σ is near criticical point =⇒Innaccuracy
Use Moore Penrose Inverse
Use diagonal matrix
Use Identity
M. Senneret, Y. Malevergne, P. Abry, G. Perrin, L. Ja�rès Covariance and Precision for Portfolio Selection
![Page 23: A Comparative Study of Covariance and Precision Matrix ... · Motivation Setup Results Summary A Comparative Study of Covariance and Precision Matrix Estimators for Portfolio Selection](https://reader034.fdocuments.in/reader034/viewer/2022043014/5fafca008eff9505ca481637/html5/thumbnails/23.jpg)
MotivationSetup
ResultsSummary
Assets & OptimizationsDirect EstimatesFactor ModelsShrinkageSparsity
Direct Estimates
if n < p , then Σ is not full rank =⇒ No inversion
if n ' p , then Σ is near criticical point =⇒Innaccuracy
Use Moore Penrose Inverse
Use diagonal matrix
Use Identity
M. Senneret, Y. Malevergne, P. Abry, G. Perrin, L. Ja�rès Covariance and Precision for Portfolio Selection
![Page 24: A Comparative Study of Covariance and Precision Matrix ... · Motivation Setup Results Summary A Comparative Study of Covariance and Precision Matrix Estimators for Portfolio Selection](https://reader034.fdocuments.in/reader034/viewer/2022043014/5fafca008eff9505ca481637/html5/thumbnails/24.jpg)
MotivationSetup
ResultsSummary
Assets & OptimizationsDirect EstimatesFactor ModelsShrinkageSparsity
Direct Estimates
if n < p , then Σ is not full rank =⇒ No inversion
if n ' p , then Σ is near criticical point =⇒Innaccuracy
Use Moore Penrose Inverse
Use diagonal matrix
Use Identity
M. Senneret, Y. Malevergne, P. Abry, G. Perrin, L. Ja�rès Covariance and Precision for Portfolio Selection
![Page 25: A Comparative Study of Covariance and Precision Matrix ... · Motivation Setup Results Summary A Comparative Study of Covariance and Precision Matrix Estimators for Portfolio Selection](https://reader034.fdocuments.in/reader034/viewer/2022043014/5fafca008eff9505ca481637/html5/thumbnails/25.jpg)
MotivationSetup
ResultsSummary
Assets & OptimizationsDirect EstimatesFactor ModelsShrinkageSparsity
Direct Estimates
if n < p , then Σ is not full rank =⇒ No inversion
if n ' p , then Σ is near criticical point =⇒Innaccuracy
Use Moore Penrose Inverse
Use diagonal matrix
Use Identity
M. Senneret, Y. Malevergne, P. Abry, G. Perrin, L. Ja�rès Covariance and Precision for Portfolio Selection
![Page 26: A Comparative Study of Covariance and Precision Matrix ... · Motivation Setup Results Summary A Comparative Study of Covariance and Precision Matrix Estimators for Portfolio Selection](https://reader034.fdocuments.in/reader034/viewer/2022043014/5fafca008eff9505ca481637/html5/thumbnails/26.jpg)
MotivationSetup
ResultsSummary
Assets & OptimizationsDirect EstimatesFactor ModelsShrinkageSparsity
Outline
1 Motivation
2 Setup
Assets & Optimizations
Direct Estimates
Factor Models
Shrinkage
Sparsity
3 Results
Volatility Control
Sharpe Control
Concentration Control
M. Senneret, Y. Malevergne, P. Abry, G. Perrin, L. Ja�rès Covariance and Precision for Portfolio Selection
![Page 27: A Comparative Study of Covariance and Precision Matrix ... · Motivation Setup Results Summary A Comparative Study of Covariance and Precision Matrix Estimators for Portfolio Selection](https://reader034.fdocuments.in/reader034/viewer/2022043014/5fafca008eff9505ca481637/html5/thumbnails/27.jpg)
MotivationSetup
ResultsSummary
Assets & OptimizationsDirect EstimatesFactor ModelsShrinkageSparsity
Factor Models
Full rank estimate when n < p
Easy and reliable estimate of Θ
Noise Reduction
Introduction of information via exogenous factor
Two ways : exogenous model (Sharpe 1963, Fama French
1992), endogenous model (PCA)
M. Senneret, Y. Malevergne, P. Abry, G. Perrin, L. Ja�rès Covariance and Precision for Portfolio Selection
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MotivationSetup
ResultsSummary
Assets & OptimizationsDirect EstimatesFactor ModelsShrinkageSparsity
Factor Models
Full rank estimate when n < p
Easy and reliable estimate of Θ
Noise Reduction
Introduction of information via exogenous factor
Two ways : exogenous model (Sharpe 1963, Fama French
1992), endogenous model (PCA)
M. Senneret, Y. Malevergne, P. Abry, G. Perrin, L. Ja�rès Covariance and Precision for Portfolio Selection
![Page 29: A Comparative Study of Covariance and Precision Matrix ... · Motivation Setup Results Summary A Comparative Study of Covariance and Precision Matrix Estimators for Portfolio Selection](https://reader034.fdocuments.in/reader034/viewer/2022043014/5fafca008eff9505ca481637/html5/thumbnails/29.jpg)
MotivationSetup
ResultsSummary
Assets & OptimizationsDirect EstimatesFactor ModelsShrinkageSparsity
Factor Models
Full rank estimate when n < p
Easy and reliable estimate of Θ
Noise Reduction
Introduction of information via exogenous factor
Two ways : exogenous model (Sharpe 1963, Fama French
1992), endogenous model (PCA)
M. Senneret, Y. Malevergne, P. Abry, G. Perrin, L. Ja�rès Covariance and Precision for Portfolio Selection
![Page 30: A Comparative Study of Covariance and Precision Matrix ... · Motivation Setup Results Summary A Comparative Study of Covariance and Precision Matrix Estimators for Portfolio Selection](https://reader034.fdocuments.in/reader034/viewer/2022043014/5fafca008eff9505ca481637/html5/thumbnails/30.jpg)
MotivationSetup
ResultsSummary
Assets & OptimizationsDirect EstimatesFactor ModelsShrinkageSparsity
Factor Models
Full rank estimate when n < p
Easy and reliable estimate of Θ
Noise Reduction
Introduction of information via exogenous factor
Two ways : exogenous model (Sharpe 1963, Fama French
1992), endogenous model (PCA)
M. Senneret, Y. Malevergne, P. Abry, G. Perrin, L. Ja�rès Covariance and Precision for Portfolio Selection
![Page 31: A Comparative Study of Covariance and Precision Matrix ... · Motivation Setup Results Summary A Comparative Study of Covariance and Precision Matrix Estimators for Portfolio Selection](https://reader034.fdocuments.in/reader034/viewer/2022043014/5fafca008eff9505ca481637/html5/thumbnails/31.jpg)
MotivationSetup
ResultsSummary
Assets & OptimizationsDirect EstimatesFactor ModelsShrinkageSparsity
Factor Models
Full rank estimate when n < p
Easy and reliable estimate of Θ
Noise Reduction
Introduction of information via exogenous factor
Two ways : exogenous model (Sharpe 1963, Fama French
1992), endogenous model (PCA)
M. Senneret, Y. Malevergne, P. Abry, G. Perrin, L. Ja�rès Covariance and Precision for Portfolio Selection
![Page 32: A Comparative Study of Covariance and Precision Matrix ... · Motivation Setup Results Summary A Comparative Study of Covariance and Precision Matrix Estimators for Portfolio Selection](https://reader034.fdocuments.in/reader034/viewer/2022043014/5fafca008eff9505ca481637/html5/thumbnails/32.jpg)
MotivationSetup
ResultsSummary
Assets & OptimizationsDirect EstimatesFactor ModelsShrinkageSparsity
Outline
1 Motivation
2 Setup
Assets & Optimizations
Direct Estimates
Factor Models
Shrinkage
Sparsity
3 Results
Volatility Control
Sharpe Control
Concentration Control
M. Senneret, Y. Malevergne, P. Abry, G. Perrin, L. Ja�rès Covariance and Precision for Portfolio Selection
![Page 33: A Comparative Study of Covariance and Precision Matrix ... · Motivation Setup Results Summary A Comparative Study of Covariance and Precision Matrix Estimators for Portfolio Selection](https://reader034.fdocuments.in/reader034/viewer/2022043014/5fafca008eff9505ca481637/html5/thumbnails/33.jpg)
MotivationSetup
ResultsSummary
Assets & OptimizationsDirect EstimatesFactor ModelsShrinkageSparsity
Shrinkage Approach
Mix between sample matrix and model matrix, nearest to the
true matrix (Ledoit and Wolf (2004)).
Σ
SnρT
ν+(1-ρ)S
nT
ν
Shrink with Identity or diagonal
Shrink to precision Θ
Sn singular =⇒ hard problem, moments of the Moore Penrose
inverse are unknown
M. Senneret, Y. Malevergne, P. Abry, G. Perrin, L. Ja�rès Covariance and Precision for Portfolio Selection
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MotivationSetup
ResultsSummary
Assets & OptimizationsDirect EstimatesFactor ModelsShrinkageSparsity
Shrinkage Approach
Mix between sample matrix and model matrix, nearest to the
true matrix (Ledoit and Wolf (2004)).
Σ
SnρT
ν+(1-ρ)S
nT
ν
Shrink with Identity or diagonal
Shrink to precision Θ
Sn singular =⇒ hard problem, moments of the Moore Penrose
inverse are unknown
M. Senneret, Y. Malevergne, P. Abry, G. Perrin, L. Ja�rès Covariance and Precision for Portfolio Selection
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MotivationSetup
ResultsSummary
Assets & OptimizationsDirect EstimatesFactor ModelsShrinkageSparsity
Shrinkage Approach
Mix between sample matrix and model matrix, nearest to the
true matrix (Ledoit and Wolf (2004)).
Σ
SnρT
ν+(1-ρ)S
nT
ν
Shrink with Identity or diagonal
Shrink to precision Θ
Sn singular =⇒ hard problem, moments of the Moore Penrose
inverse are unknown
M. Senneret, Y. Malevergne, P. Abry, G. Perrin, L. Ja�rès Covariance and Precision for Portfolio Selection
![Page 36: A Comparative Study of Covariance and Precision Matrix ... · Motivation Setup Results Summary A Comparative Study of Covariance and Precision Matrix Estimators for Portfolio Selection](https://reader034.fdocuments.in/reader034/viewer/2022043014/5fafca008eff9505ca481637/html5/thumbnails/36.jpg)
MotivationSetup
ResultsSummary
Assets & OptimizationsDirect EstimatesFactor ModelsShrinkageSparsity
Shrinkage Approach
Mix between sample matrix and model matrix, nearest to the
true matrix (Ledoit and Wolf (2004)).
Σ
SnρT
ν+(1-ρ)S
nT
ν
Shrink with Identity or diagonal
Shrink to precision Θ
Sn singular =⇒ hard problem, moments of the Moore Penrose
inverse are unknown
M. Senneret, Y. Malevergne, P. Abry, G. Perrin, L. Ja�rès Covariance and Precision for Portfolio Selection
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MotivationSetup
ResultsSummary
Assets & OptimizationsDirect EstimatesFactor ModelsShrinkageSparsity
Outline
1 Motivation
2 Setup
Assets & Optimizations
Direct Estimates
Factor Models
Shrinkage
Sparsity
3 Results
Volatility Control
Sharpe Control
Concentration Control
M. Senneret, Y. Malevergne, P. Abry, G. Perrin, L. Ja�rès Covariance and Precision for Portfolio Selection
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MotivationSetup
ResultsSummary
Assets & OptimizationsDirect EstimatesFactor ModelsShrinkageSparsity
Sparsity Justi�cation
Principle of parsimony =⇒ Occam's razor
a priori theoretical justi�cation:
assets of di�erent class are weakly related with others, e.g.bonds/stockswithin a class, conditional correlation may be weak
Statistical justi�cation: Screening e�ect =⇒ o�-diagonal
terms can be statistically indistinguishable from 0
Which coe�cients must be forced to 0?
M. Senneret, Y. Malevergne, P. Abry, G. Perrin, L. Ja�rès Covariance and Precision for Portfolio Selection
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MotivationSetup
ResultsSummary
Assets & OptimizationsDirect EstimatesFactor ModelsShrinkageSparsity
Sparsity Justi�cation
Principle of parsimony =⇒ Occam's razor
a priori theoretical justi�cation:
assets of di�erent class are weakly related with others, e.g.bonds/stockswithin a class, conditional correlation may be weak
Statistical justi�cation: Screening e�ect =⇒ o�-diagonal
terms can be statistically indistinguishable from 0
Which coe�cients must be forced to 0?
M. Senneret, Y. Malevergne, P. Abry, G. Perrin, L. Ja�rès Covariance and Precision for Portfolio Selection
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MotivationSetup
ResultsSummary
Assets & OptimizationsDirect EstimatesFactor ModelsShrinkageSparsity
Sparsity Justi�cation
Principle of parsimony =⇒ Occam's razor
a priori theoretical justi�cation:
assets of di�erent class are weakly related with others, e.g.bonds/stockswithin a class, conditional correlation may be weak
Statistical justi�cation: Screening e�ect =⇒ o�-diagonal
terms can be statistically indistinguishable from 0
Which coe�cients must be forced to 0?
M. Senneret, Y. Malevergne, P. Abry, G. Perrin, L. Ja�rès Covariance and Precision for Portfolio Selection
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MotivationSetup
ResultsSummary
Assets & OptimizationsDirect EstimatesFactor ModelsShrinkageSparsity
Sparsity Justi�cation
Principle of parsimony =⇒ Occam's razor
a priori theoretical justi�cation:
assets of di�erent class are weakly related with others, e.g.bonds/stockswithin a class, conditional correlation may be weak
Statistical justi�cation: Screening e�ect =⇒ o�-diagonal
terms can be statistically indistinguishable from 0
Which coe�cients must be forced to 0?
M. Senneret, Y. Malevergne, P. Abry, G. Perrin, L. Ja�rès Covariance and Precision for Portfolio Selection
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MotivationSetup
ResultsSummary
Assets & OptimizationsDirect EstimatesFactor ModelsShrinkageSparsity
Principles
Trade-o� between data �delity and promoting sparsity
Most common methods act on precision (Boyd et al. 2010 ):
Θ̂ = argminΘ
Tr(SnΘ)− log det Θ + λ ||Θ||1
l1 norm acts as a surrogate to the counting of non-zero terms
New method acts directly on covariance, but is non convex
(Bien & Tibshirani 2012)
Σ̂ = argminΣ
Tr(SnΣ−1) + log det Σ + λ ||Σ||1
M. Senneret, Y. Malevergne, P. Abry, G. Perrin, L. Ja�rès Covariance and Precision for Portfolio Selection
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MotivationSetup
ResultsSummary
Assets & OptimizationsDirect EstimatesFactor ModelsShrinkageSparsity
Principles
Trade-o� between data �delity and promoting sparsity
Most common methods act on precision (Boyd et al. 2010 ):
Θ̂ = argminΘ
Tr(SnΘ)− log det Θ + λ ||Θ||1
l1 norm acts as a surrogate to the counting of non-zero terms
New method acts directly on covariance, but is non convex
(Bien & Tibshirani 2012)
Σ̂ = argminΣ
Tr(SnΣ−1) + log det Σ + λ ||Σ||1
M. Senneret, Y. Malevergne, P. Abry, G. Perrin, L. Ja�rès Covariance and Precision for Portfolio Selection
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MotivationSetup
ResultsSummary
Assets & OptimizationsDirect EstimatesFactor ModelsShrinkageSparsity
Principles
Trade-o� between data �delity and promoting sparsity
Most common methods act on precision (Boyd et al. 2010 ):
Θ̂ = argminΘ
Tr(SnΘ)− log det Θ + λ ||Θ||1
l1 norm acts as a surrogate to the counting of non-zero terms
New method acts directly on covariance, but is non convex
(Bien & Tibshirani 2012)
Σ̂ = argminΣ
Tr(SnΣ−1) + log det Σ + λ ||Σ||1
M. Senneret, Y. Malevergne, P. Abry, G. Perrin, L. Ja�rès Covariance and Precision for Portfolio Selection
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MotivationSetup
ResultsSummary
Volatility ControlSharpe ControlConcentration Control
Outline
1 Motivation
2 Setup
Assets & Optimizations
Direct Estimates
Factor Models
Shrinkage
Sparsity
3 Results
Volatility Control
Sharpe Control
Concentration Control
M. Senneret, Y. Malevergne, P. Abry, G. Perrin, L. Ja�rès Covariance and Precision for Portfolio Selection
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MotivationSetup
ResultsSummary
Volatility ControlSharpe ControlConcentration Control
Volatility - Input: Covariance
Volatility 150 200 350 150 200 350
Unconstrained Constrained
Basic approachesSample 32.44 43.48 10.42 13.72 14.87 10.26
Identity 20.88 20.88 20.88 20.88 14.87 20.88
Diagonal 16.79 16.93 17.44 16.82 20.88 17.46
Factor modelsACP 1F 9.03 9.16 9.93 9.88 9.93 10.43
ACP 2F 9.05 9.24 10.01 9.82 9.89 10.41
Market Model 16.43 16.60 17.17 16.47 16.65 17.2
ShrinkageIdentity 8.97 9.04 9.18 9.82 9.86 10.25Diagonal 9.73 8.48 8.81 9.73 9.84 10.25
Sparsity 16.81 16.95 13.99 14.99
M. Senneret, Y. Malevergne, P. Abry, G. Perrin, L. Ja�rès Covariance and Precision for Portfolio Selection
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MotivationSetup
ResultsSummary
Volatility ControlSharpe ControlConcentration Control
Volatility - Input: Precision
Volatility 150 200 350 150 200 350
Unconstrained Constrained
Basic approachesSample 32.44 43.48 10.42 13.72 14.87 10.26
Identity 20.88 20.88 20.88 20.88 14.87 20.88
Diagonal 16.79 16.93 17.44 16.82 20.88 17.46
Factor modelsACP 1F 9.03 9.16 9.93 9.88 9.93 10.43
ACP 2F 9.05 9.24 10.01 9.82 9.89 10.41Market Model 17.43 17.34 17.73 17.45 17.36 17.74
ShrinkageIdentity - - 20.60 - - 20.71
Diagonal - - 19.28 - - 19.3
Sparsity 7.81 16.63 10.1 10.3 12.34 11.64
M. Senneret, Y. Malevergne, P. Abry, G. Perrin, L. Ja�rès Covariance and Precision for Portfolio Selection
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MotivationSetup
ResultsSummary
Volatility ControlSharpe ControlConcentration Control
Outline
1 Motivation
2 Setup
Assets & Optimizations
Direct Estimates
Factor Models
Shrinkage
Sparsity
3 Results
Volatility Control
Sharpe Control
Concentration Control
M. Senneret, Y. Malevergne, P. Abry, G. Perrin, L. Ja�rès Covariance and Precision for Portfolio Selection
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MotivationSetup
ResultsSummary
Volatility ControlSharpe ControlConcentration Control
Sharpe - Input: Covariance
Sharpe 150 200 350 150 200 350
Unconstrained Constrained
Basic approachesSample 0.72 0.23 0.8 0.84 0.94 0.86
Identity 0.72 0.72 0.72 0.72 0.72 0.72
Diagonal 0.67 0.74 0.71 0.67 0.74 0.71
Factor modelsACP 1F 0.93 1.02 0.99 0.97 1.03 0.85
ACP 2F 0.96 1.03 1.02 0.92 1.00 0.81
Market Model 0.68 0.74 0.71 0.68 0.75 0.71
ShrinkageIdentity 1.14 1.11 1.11 1.15 1.18 0.95Diagonal 1.05 1.08 1.09 1.05 1.09 0.86
Sparsity 0.67 0.74 0.82 0.83
M. Senneret, Y. Malevergne, P. Abry, G. Perrin, L. Ja�rès Covariance and Precision for Portfolio Selection
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MotivationSetup
ResultsSummary
Volatility ControlSharpe ControlConcentration Control
Sharpe - Input: Precision
Sharpe 150 200 350 150 200 350
Unconstrained Constrained
Basic approachesSample 0.72 0.23 0.8 0.84 0.94 0.86
Identity 0.72 0.72 0.72 0.72 0.72 0.72
Diagonal 0.67 0.74 0.71 0.67 0.74 0.71
Factor modelsACP 1F 0.93 1.02 0.99 0.97 1.03 0.85
ACP 2F 0.96 1.03 1.02 0.92 1.00 0.81
Market Model 0.67 0.73 0.72 0.67 0.73 0.72
ShrinkageIdentity - - 0.71 - - 0.71
Diagonal - - 0.62 - - 0.62
Sparsity 1.33 1.05 1.11 0.89 1 0.95
M. Senneret, Y. Malevergne, P. Abry, G. Perrin, L. Ja�rès Covariance and Precision for Portfolio Selection
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MotivationSetup
ResultsSummary
Volatility ControlSharpe ControlConcentration Control
Sharpe - Input: Precision
Sharpe 150 200 350 150 200 350
Unconstrained Constrained
Basic approachesSample 0.72 0.23 0.8 0.84 0.94 0.86
Identity 0.72 0.72 0.72 0.72 0.72 0.72
Diagonal 0.67 0.74 0.71 0.67 0.74 0.71
Factor modelsACP 1F 0.93 1.02 0.99 0.97 1.03 0.85
ACP 2F 0.96 1.03 1.02 0.92 1.00 0.81
Market Model 0.67 0.73 0.72 0.67 0.73 0.72
ShrinkageIdentity - - 0.71 - - 0.71
Diagonal - - 0.62 - - 0.62
Sparsity 1.33 1.05 1.11 0.89 1 0.95
M. Senneret, Y. Malevergne, P. Abry, G. Perrin, L. Ja�rès Covariance and Precision for Portfolio Selection
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MotivationSetup
ResultsSummary
Volatility ControlSharpe ControlConcentration Control
Outline
1 Motivation
2 Setup
Assets & Optimizations
Direct Estimates
Factor Models
Shrinkage
Sparsity
3 Results
Volatility Control
Sharpe Control
Concentration Control
M. Senneret, Y. Malevergne, P. Abry, G. Perrin, L. Ja�rès Covariance and Precision for Portfolio Selection
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MotivationSetup
ResultsSummary
Volatility ControlSharpe ControlConcentration Control
Unconstrained Optimization
Sharpe Turnover Her�ndahl−1 Short int.
150
Sample 0.72 27.69 1.62 -16.52
Cov-Id Shrinkage 1.14 2.71 12.51 -3.46
Sparsity Precision 1.33 0.76 14.94 -1.18
200Sample 0.23 23.57 0.5 -10.89
Cov-Id Shrinkage 1.11 1.05 10.07 -1.52
Sparsity Precision 1.05 0.08 48.48 -0.08
350Sample 0.23 23.57 0.5 -10.89
Cov-Id Shrinkage 1.11 1.77 7.98 -3.69
Sparsity Precision 1.11 0.21 30.11 -0.48
M. Senneret, Y. Malevergne, P. Abry, G. Perrin, L. Ja�rès Covariance and Precision for Portfolio Selection
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MotivationSetup
ResultsSummary
Summary
Con�rmation of the relevance of shrinkage when focusing on
covariance-based optimization.
Con�rmation of the positive impact of constraints on badly
estimated covariance matrix.
Negative impact of constraints when good estimate is
available.
Precision-based optimization is sightly better than covariance
when unconstrained.
Precision-based optimization leads to much more stable
portfolios.
M. Senneret, Y. Malevergne, P. Abry, G. Perrin, L. Ja�rès Covariance and Precision for Portfolio Selection
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MotivationSetup
ResultsSummary
Summary
Con�rmation of the relevance of shrinkage when focusing on
covariance-based optimization.
Con�rmation of the positive impact of constraints on badly
estimated covariance matrix.
Negative impact of constraints when good estimate is
available.
Precision-based optimization is sightly better than covariance
when unconstrained.
Precision-based optimization leads to much more stable
portfolios.
M. Senneret, Y. Malevergne, P. Abry, G. Perrin, L. Ja�rès Covariance and Precision for Portfolio Selection
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MotivationSetup
ResultsSummary
Summary
Con�rmation of the relevance of shrinkage when focusing on
covariance-based optimization.
Con�rmation of the positive impact of constraints on badly
estimated covariance matrix.
Negative impact of constraints when good estimate is
available.
Precision-based optimization is sightly better than covariance
when unconstrained.
Precision-based optimization leads to much more stable
portfolios.
M. Senneret, Y. Malevergne, P. Abry, G. Perrin, L. Ja�rès Covariance and Precision for Portfolio Selection
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MotivationSetup
ResultsSummary
Summary
Con�rmation of the relevance of shrinkage when focusing on
covariance-based optimization.
Con�rmation of the positive impact of constraints on badly
estimated covariance matrix.
Negative impact of constraints when good estimate is
available.
Precision-based optimization is sightly better than covariance
when unconstrained.
Precision-based optimization leads to much more stable
portfolios.
M. Senneret, Y. Malevergne, P. Abry, G. Perrin, L. Ja�rès Covariance and Precision for Portfolio Selection
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AppendixConstraints
M. Senneret, Y. Malevergne, P. Abry, G. Perrin, L. Ja�rès Covariance and Precision for Portfolio Selection
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AppendixConstraints
Regularization by constraint
KKT2Σw + λ ·1p−ν = 0
1′w −1 = 0
Diag(ν) w = 0
w ,ν ≥ 0
Equivalent to
2[Σ + Diag(ν)− 1
2· (1pν
′+ ν1′p)]w + λ ·1p = 0
Constraints impose an e�ective decay on correlation
M. Senneret, Y. Malevergne, P. Abry, G. Perrin, L. Ja�rès Covariance and Precision for Portfolio Selection